Mathematics
http://hdl.handle.net/2429/19748
2015-05-28T21:18:24ZAn enhanced pseudo-3D model for hydraulic fracturing accounting for viscous height growth, non-local elasticity, and lateral toughness (Issued:2015-02-27)
http://hdl.handle.net/2429/52289
The goal of this paper is to develop an enhanced pseudo-3D (P3D) model for hydraulic fracturing (HF), whose predictions are more accurate compared to that of the original P3D model, but which requires significantly less computational resources than a fully planar HF simulator. We show that the lack of viscous resistance in the height growth and the local approximation in the computation of elastic interactions, which precludes the incorporation of lateral toughness, are the primary weaknesses of the original P3D model that considers symmetric stress barriers and no leak-off. To account for the viscous resistance, an apparent fracture toughness is introduced. The apparent toughness is calibrated using a one-dimensional HF model resulting in an approximate expression that captures all regimes of propagation. To incorporate non-local elastic interactions, the fracture opening in every vertical cross-section is approximated by a plane-strain solution, and then the 2D elasticity interaction integral is evaluated. To increase the computational efficiency, this 2D integral is further approximated by two one-dimensional integrals. The use of non-local elasticity allows us to include the asymptotic solution in the tip element, and, in particular, to include the effect of lateral fracture toughness. To further increase the accuracy of the P3D model, the flat fracture tip is replaced by its curved counterpart. This also permits us to capture radial behaviour at early times before the fracture has reached the stress barriers. To evaluate the accuracy of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar HF simulator, which is able to capture viscous, toughness, and intermediate propagation regimes. It is shown that the enhanced P3D model is able to approximate the propagation of hydraulic fractures accurately for various regimes of propagation, as well as for different fracture aspect ratios.
2015-02-27T00:00:00ZOn the Hilbert-Pólya and Pair Correlation Conjectures (Issued:2014-01-01)
http://hdl.handle.net/2429/50314
The Hilbert-Pólya Conjecture supposes that there exists an operator in a Hilbert space whose eigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would very likely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s) have real part 1/2. In this thesis we summarize work by Berry, Keating and others in constructing such an operator. Although the work so far has not yet yielded such an operator, some have been found that have properties very close to what is desired. We also summarize a (partially proven) conjecture by Montgomery that motivates the search for this operator. He conjectures that the pair correlation function for the spacing between the imaginary parts of the Riemann zeroes is the same as the correlation function for the spacing between eigenvalues of random Gaussian unitary matrices.
2014-01-01T00:00:00ZSlurry flow, gravitational settling, and a proppant transport model for hydraulic fractures (Issued:2014-08-13)
http://hdl.handle.net/2429/50000
The goal of this study is to analyze the steady flow of a Newtonian fluid mixed with spherical particles in a channel based on a continuum model, where the constitutive behaviour of the slurry is approximated
by an empirical formula. In order to account for the gravitational settling of particles, two-dimensional
flow needs to be considered as the pressure gradient and gravity may not always be collinear. It is shown
that the problem under consideration features a boundary layer, whose size is on the order of the particle
radius. The expressions for both the outer (i.e. outside the boundary layer) and inner (i.e. within the
boundary layer) solutions are obtained in terms of the particle concentration, particle velocity, and fluid velocity. Unfortunately, these solutions require numerical solution of an integral equation, depend on the ratio between the pressure gradient and the gravity force, and the orientation of the pressure gradient relative to the gravity. Consequently, the development of a proppant transport model for hydraulic fracturing based on these results is not practical. For this reason, an approximate solution is introduced, where the effect of gravity is accounted for in an approximate fashion, reducing the complexity of the slurry flow solution. To validate the use of this approximation, the error is estimated for different regimes of flow. The approximate solution is then used to calculate the expressions for the slurry flux and the proppant flux, which are the basis for a model that can be used to account for proppant transport with gravitational settling in a fully coupled hydraulic fracturing simulator.
2014-08-13T00:00:00ZA new paradigm for proppant schedule design (Issued:2014-02-27)
http://hdl.handle.net/2429/46110
This study introduces a novel methodology for the design of the proppant pumping schedule for a hydraulic fracture, in which the fi nal proppant distribution along the crack is prescribed. The method is based on the assumption that the particles have relatively small impact on the fracture propagation, unless they reach the tip region. This makes it possible to relate the proppant velocity to the clear fluid velocity inside the fracture, which is calculated assuming no proppant. Having the history of the clear
fluid velocity distribution, the prospective proppant motion can be computed. Then, volume
balance is used to relate the final concentration at some point inside the fracture to the corresponding input concentration at a speci fic time instant, which helps to avoid solving an inverse problem. One exceptional feature of the approach lies in the fact that it is applicable to multiple fracture geometries and can be implemented using various hydraulic fracturing simulators. To verify the technique, two fracture geometries are considered - Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). It is shown that the developed approach is capable of properly estimating the pumping schedule for both geometries. In particular, the proppant placement along the fracture at the end of the pumping
period, calculated according to the adopted proppant transport model, shows close agreement with the design distribution. The comparison with Nolte's scheduling scheme shows that the latter is not always accurate, and cannot capture the essential di fferences between the schedules for the fracture geometries considered.
2014-02-27T00:00:00Z